Hybrid Identities and Hybrid Equational Logic

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  • Math. Log. Quart. 41 (1995) 190 - 196

    Mathematical Logic Quarterly

    @ Johann Ambrosius Barth 1995

    Hybrid Identities and Hybrid Equational Logic

    Klaus Denecke

    Institut fur Mathematik, Universitat Potsdam, Am Neuen Palais, D-14415 Potsdam, Germany)

    Abstract. Hybrid identities are sentences in a special second order language with identity. The model classes of sets of hybrid identities are called hybrid solid varieties. We give a Birkhoff-type-characterization of hybrid solid varieties and develop a hybrid equational logic.

    Mathematics Subject Classification: 08B05, 03B15, 03C05. Keywords: Hyperidentity, Hybrid identity, Hybrid solid variety, Hybrid equational theory.

    1 Introduction

    An identity t R t of terms of any type T is called a hybr id iden t i t y for a universal algebra A = ( A ; (ft)iE1) if t x t holds identically for every choice of n-ary term op- erations to represent some n-ary operation symbols occuring in t and t. For example, the equation

    (1) F ( x A Y, 2) M F ( r , 2) A F(Y, Z), where F is a binary operation symbol is a hybrid identity for every distributive lattice. Indeed, any binary term operation of a distributive lattice is induced by one of the following binary terms: 2, y, xAy, x v y . Replacing F in (1) by one of these binary term operations we get an identity in any distributive lattice. An identity t M 2 is called a hyper iden t i t y for A if t M t holds identically in A whenever all operation symbols occuring in t and t are replaced by any term operation of A of the appropriate arity. So, hybrid identities are between identities and hyperidentities. The concept of a hybrid identity was introduced by SCHWEIGERT in [4]. Hyperidentities and hybrid identities may be considered as specific sentences in a second order language with equality. In the sentence (1) the symbol F stands for a binary operation defined on A and we bind the interpretation of F to binary term operations of the algebra A.

    If V is a variety of algebras of a given type T , then V is called hybr id so l id if each of its identities is a hybrid identity. It turns out that hybrid solid varieties are just model classes of sets of hybrid identities. We give a Birkhoff type characterization of hybrid solid varieties and prove that all hybrid solid varieties of a given type T form a lattice which is a sublattice of the lattice of all varieties of type T. Then we develop a hybrid equational logic (a fragment of second order logic) containing the hybrid substitution rule as an additional rule and prove completeness.

    l)e-rnail: KdeneckeQhp.rz.uni-potsdam.de

  • Hybrid Identities and Hybrid Equational Logic 191

    2 Hybrid substitutions

    We will give a more precise definition of a hybrid identity using the concept of a hybrid substitution. We fix a type r = {ni I i E I, ni > 0 for all i E I}, and operation symbols {f* I i E I}, where fi is n,-ary. Let W7(X) be the set of all terms of type r over some fixed alphabet X, and let Alg(r) be the class of all algebras of type r.

    D e f i n i t i o n 2.1. Let I beasubset o f I . Thenan I/-hybridsubstitution o f t y p e r , (for short, an 1-hybrid substitution), is a mapping a : {f, I i E I } - W7(X) which assigns to every n,-ary operation symbol an n;-ary term, (i. e. a term constructed from the variables zl,. . .,I,,,) such that u(fi) = fi(z1,. . . ,zn,) for all i E 1.

    Clearly, every 1-hybrid substitution can be extended to the terms by the following inductive definition :

    (i) &[z] := z for any variable in the alphabet X, (ii) e[f,(tl,. . . ,tn,)l := ~I(fi)(eI[tl]~. . .,&I[tn,l) for f , ( t l , . . . , i n , ) E w ~ ( x ) .

    If t w t is an equation, then we denote by Z:[t x t] the set

    uI( j , ) = f , ( q , . . . ,zn,) for all i E 11. {b[t] x (i[t] I ul : {f, I i E I} - W T ( X ) and

    Let A = ( A ; ( f t ) i E i ) be an algebra of type 7. Then we define

    .[dl := ( A ; (uI(fi)A)iEl), Z:[d] := {u[d] I (TI : {fi 1 i E 1) - W 7 ( X ) and

    u( f i ) = fi(z1,. . . ,zn,) for all i E 11. For a set C of equations consisting of terms of type r and for a class K of algebras of this type we put

    Z:[C] = U { Z I [ t x t] I t x 2 E C}, E[K] = U{z:[Jt] I A E K}. P r o p o s i t i o n 2.2. For all sets 0 c I c I , Zz is a closure operator on sets of

    equations C and on classes of algebras K of type r , i. e.,

    (i) c z:[cI, (i) K g Z:[K], (ii) C g C 3 Z.[C] Z1[C], (ii) K 5 K +- ZII[K] g ZI[K],

    (iii) 81 [c - I [C]] = Z[C], (iii) Z[S:[K]] = ZI[K].

    P r o o f . Note that I-hybrid substitutions are hypersubstitutions in the sense of [l], i . e. mappings u : {fi I i E I} - W 7 ( X ) which assign to every nj-ary operation symbol an ni-ary term. We define the composition of two I-hybrid substitutions by 01 oh u g ( f i ) := &1[ug(fi)] for all i E 1. Further put uid : fi H fi(z1,. . . , z,,). Since b i d is an 1-hybrid substitution, (i) and (1) are clear. The propositions (ii) and (ii) are also obvious. Let A E K and let u, a be I/-hybrid substitutions. Assume that u[u[d]] = (A; ( ~ ( f ~ ) ~ ~ [ ~ ] ) i ~ ~ E Z[EI[Ii]]. Since the composition of 1-hybrid substitutions is again an I/-hybrid substitution, the algebra u[u[d]] is an element of Z:[K], and together with E[K] 5 E[E1[K]] we have (iii). The proof of (iii) is similar and straightforward. 0

  • 192 Klaus Denecke

    P r o p o s i t i o n 2.3. For sets I' G I" G I , for any set C of equations and for any class K of algebras of type r we have:

    (i) E"[C] 2 Z'"[C] and (ii) 3"[K] 2 Z:"'[K]. 0

    3 1'-hybrid identities

    Using I/-hybrid substitutions we can give a more precise definition of the 1'-hybrid identities.

    D e f i n i t i o n 3.1. Let d E Alg(r), let { f, I i E I} be a set of operation symbols, and let I' be a subset of I . Then the identity t w t ' , where t ,t ' are terms of type r , is called an I'-hybrid identity of type T in A if # [ t ] w 6."[t'] are identities for every I/-hybrid substitution cd'. If K is a class of algebras of type r , then the identity t M t' is called an 1'-hybrid identity in K if it is an I/-hybrid identity in every algebra of K.

    Note that @-hybrid identities are hyperidentities and that I-hybrid identities are ordinary identities.

    For a class K of algebras of type T and for a set C of identities of this type we fix the following notations: Id K denotes the class of all identities of K , HId'IK denotes the class of all 1'-hybrid identities of K , Mod C denotes {d E Alg(r) 1 A satisfies C} (the variety defined by C), Var K denotes ModId K (the variety generated by K) , HMod'lC denotes {A E Alg(7) I A satisfies every equation from C as an 1'-hybrid identity} (the I/-hybrid equational class defined by C), and HVarI'K denotes HMod"H1d"K (the I/-hybrid equational class of type T defined by H1d"K).

    By definition every I'-hybrid identity is an identity. Very natural there arises the problem to find algebras or varieties for which every identity is an I/-hybrid identity.

    D e f i n i t i o n 3.2. Let V be a variety of type r . Then V is called I'-hybrid solid if E"[V] = V. (If I' = 0, then V is called solid variety of type r.)

    T h e o r e m 3.3. Let K c Alg(r) be a variety. Then the following conditions are equivalent:

    (i) K i s an I'-hybrid equational class, (ii) K is I/-hybrid solid, (iii) Id K = HId'lK, i . e. every identity of K is an 1'-hybrid identity, (iv) E:"[Id K] = Id K , i. e. Id K i s closed under If-hybrid substitutions.

    P r o o f . The first step is to prove that for every algebra d E K there holds

    (2) This is true by definition of Z". The inclusion Z:"[t w t ' ] fact that t M t' is an It-hybrid identity of A, and we have

    (3) HId'IK = IdZ"[K].

    Let C be a set of equations of type r and let A E HMod."C, i. e., every equation of C is an 1'-hybrid identity in A and thus C I d d by definition

    t w t' E IdZ:"[d] E"[t w t ' ] c I d d . I d d is equivalent to the

    H1d"A. Then E:"[C]

  • Hybrid Identities and Hybrid Equational Logic 193

    of an 1-hybrid identity. This means by (2) C C IdE.[d] and thus A E ModE:[C]. Conversely, A E Mod =[El implies A E HModlC and we have

    (4) HModIC = ModE:[C].

    With C = H1dK from (3) and (4) we obtain:

    HModH1dK = Mod E[HIdK] = Mod Id Z[K],

    and therefore

    (5) HVarlK = Var Z[K].

    Now, let K be an I-hybrid equational class, i .e. K = HVarIK. Then by (5) we have = I [K] C Var B[K] = K . Together with the closure property of E we get E[K] = K and K is I/-hybrid solid.

    Next, let K be 1-hybrid solid. By definition we have =:[K] = K and further Id I{ = Id E[K] = HIdIK by (3). Therefore (iii) is satisfied.

    From Id K = H1dK by definition of an 1-hybrid identity it follows that Id K is closed under I/-hybrid substitutions. This shows (iv).

    By definition of an 1-hybrid identity the equation Id K = H1dK implies Id K = HId K and further K = Var K = ModId K = Mod E:[Id K] = Mod E[HIdK] = HModH1dK by (4). This means K = HVarIK and K is an 1-hybrid equational class. 0

    Note that the equivalence of (i) and (ii) is a Birkhoff-type-characterization of I/-hybrid equational classes. A variety is an I/-hybrid equational class if and only if it is closed under the operator E.

    4 The lattice of all 1-hybrid solid varieties

    Let C ( t ) be the lattice of d.11 varieties of type t and let S(T) be the set of all 1-hybrid solid varieties of this type. Then we have:

    T h e o r e m 4.1. S1(7) forms a sublattice o f t h e lattice L(7). P r o o f . Assume that V1, Vz E S( t ) . Then VI A VZ = V1 n V 2 , Z:[V1 ~ V Z ] g

    =:[Vi] = Vi, ( i = 1,2) , and E1 [Vlnv~] E V l n V z . Therefore, by the closure property of E, we have Z,[V1 n V,] = V1 n VZ. This means, V1 A VZ E S1(7).

    Let Id V1, Id Vz be the sets of all identities of the I/-hybrid solid varieties V1 and VZ. Then Id V1 n Id VZ E E:[IdV1 n Id VZ] because of the closure property of EI, and from IdV1 nIdV, E IdVi, ( i = 1,2), the monotony of = and Theorem 3.3(iv) it follows =[Id V1 n Id V,] g Id V1 n Id Vz , and therefore we have =:[Id V1 n Id Vz] = IdV1 n IdVz. Thus,

    V1 V V2 = Mod (IdV1 n Id Vz) = Mod Z[IdV1 n Id Vz] = HMod(IdV1 n IdVz)

    by (4) in the proof of Theorem 3.3. Since HMod(IdV1 n IdVz) is an 1-hybrid equational class, by Theorem 3.3(ii) it is 1-hybrid solid. 0

  • 194 Klaus Denecke

    Note that the proofs of Theorem 3.3 and Theorem 4.1 are very similar to the proofs of corresponding propositions on hyperequational classes and solid varieties (see [l]). Indeed, for I = 0 we get this case.

    P r o p o s i t i o n 4.2. Let {fi I i E I} be a set of operation symbols of type r and I c I c I. Then S(T) forms a sublattice of S(T) (and both are sublattices

    P r o o f . Let V be I/-hybrid solid. Then Z[V] = V. By Proposition 2.3 we have V = Z:[V] _> Z[V] _> V I m d therefore Z[V] = V. This shows that V is I-hybrid solid. Since S(r) and S ( r ) are sublattices of C ( r ) , the lattice S(r) is asublattice

    of .qT)) .

    of s ( T ) . 0 5 Hybrid equational logic

    Theorem 3.3 contains a slight modification and generalization of BIRKHOFFS char- acterization theorem for equational classes. We are asking for a generalization of the completeness theorem for equational theories.

    D e f i n i t i o n 5.1. A set C of equations of type T is called an 1-hybrid equational theorie of type r if there is a set K of algebras of type r with C = HIdlK.

    C o r o l l a r y 5.2. A n equational theory C of type T is an I-hybrid equational theory of type r ~ ~ z : [ c I = C.

    P r o o f . If C is an equational theory, then there is a class K of algebras of type r with C = Id K , i.e. K is a variety. Assume that =[El = C. Then S[IdK] = IdK for the variety K, and by Theorem 3.3 K is an I/-hybrid equational class. Conversely, assume that C is an I-hybrid equational theory of type r . Therefore by definition there is a variety K with C = HIdIK. By definition of an I-hybrid identity we have C HIdlK if and only if Z[C] c IdK, and from H1dK c HIdIK we get 3[HIdK] C IdK and further Z[Z[HIdK]] c HIdlK by Proposition 2.2 and therefore Z[HIdK] c HIdlK. This means E:[C] C. Together with Proposition

    U

    Hybrid equational logic can be done in full analogy to the ordinary equational logic of Universal Algebra. Let C be an I/-hybrid equational theory of type r and let t , t E W T ( X ) . Then we write

    2.2 we have Z[C] = C.

    C kI1 t M t

    if any algebra A of type r which satisfies every equation from C as an It-hybrid identity also satisfies t M t as an I-hybrid identity. We write

    C I - I I t R5 t

    if there is a formal deduction of t M t starting with identities in C and using the following rules of derivation (1) - (6).

    (1) 0 1 t M t for any term t E w ~ ( x ) ; (2) t M t (3) {t M t,t M ,I/) !-I t x t;

    t M t ;

  • Hybrid Identities and Hybrid Equational Logic 195

    (4) { t j M t j l I 1 5 j 5 nil tI f i ( t 1 , . . . , tn , ) M f i ( t i , . . . , t ~ , ) for every operation symbol fi (i E I ) ;

    ( 5 ) let t , t, T E W T ( X ) and let t, i? be the terms obtained from t , 2 by replac- ing every occurence of a given variable x E X by r , then t M t t f M ? (substitution rule);

    (6) t z, t tr a[t] M a[t] for every 1-hybrid substitution or (hybrid substitu- tion rule).

    We have the following completeness theorem for 1-hybrid equational logic: T h e o r e m 5.3. C k t M 2 P r o o f . Since HModrC is an I/-hybrid equational class and every identity is an

    1-hybrid identity, C k t M t is equivalent t o t M t E H1dHModC = Id HModIC. Because of HModC = ModEi[C] (see (4) in the proof of Theorem 3.3) we get t w t E Id Mod Z:[C]. This means by the completeness theorem for equational logic that t M t is derivable from E:[C] by using the derivation rules (1) - (5) for the equational logic. For hypersubstitutions E. G R A C Z Y ~ K A showed in [2] that the rule (6) commutes with the rules (1) - (5) in the sense that (6) always can be performed first. Since I/-hybrid substitutions are hypersubstitutions, the equation t M t is derivable from E:[C] by using of (1) - (5) if and only if t M t is derivable from C as

    0

    if and only if C I- t M t.

    an 1-hybrid identity by using of (1) - (6).

    6 Anexample

    Remember that a projection algebra A E Alg(r) is an algebra for which every fun- damental operation is a projection. Let P, be the class of all projection algebras of type T and let RA, be the variety generated by the class P,. The elements of RAT are called rectangular algebras of type r in [3]. We mention the following results on the variety RAT.

    L e m m a 6.1.

    (i) Let C! be the set of following equations o f type T , where i , j E I :

    (IDf,) (ABj,) (Mf.,j,) fi(fj(2119. ..yx1nj),fj(x~1,. ..,x znj),...,fj(xn,li...,xn,n,))

    fi(z,. . . , x) w 2, fi(211rx22,.. . , x n , n , ) M fi(fi(xllr...rxn,l),...,fi(Cln,r. . . , x n , n , ) ) ,

    M fj(fi(x11,. ..,x n,1),fi(212,...2n.~)r...,fi(xlnj,...,xn,nj)). Then RAT = Mod E!.

    product of two-element projection algebras. (ii) Every rectangular algebra of finite t ype r is isomorphic to a subalgebra of a direct

    0

    Let I 5 I and let k = (...